10.5 Parametric Equations

Parametric equations A third variable t (a parameter) tells us when an object is at a given point (x, y) Both x and y are functions of time (t) If f and g are continuous functions of t on an interval I then x = f(t) and y = g(t) Different parametric equations can be used to represent various speeds at which an object travels a given path

On the calculator Change MODE to PAR In y= you will now have X 1t and Y 1t Use the window to set the max and min time values as well as the max and min x and y values

Eliminating the parameter Solve one of the parametric equations for t Substitute into the other equation for t The rectangular equation should contain x and y variables only Converting from parametric to rectangular can change the range of x and y so you may need to restrict your range in the rectangular equation

Find the rectangular equation for Parametric Rectangular Since the parametric is defined only when t ≥0 and the x is always positive, you must restrict the domain of the rectangular so that x≥0

Angle parameters Solve for the trig function and use a trig identity to substitute in the values and form an equation in x and y Example: x = cosθ y = 3 sin θ sin θ= y/3 sin 2 θ + cos 2 θ=1 (y/3) 2 + x 2 = 1 Ellipse with center (0, 0)

Exponential parameters Recall that e x and ln x are inverse functions Example: x = ln 2t y = 2t 2 e x = e ln2t e x = 2t t= e x /2